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BanachTarski Paradox 
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tommyk

1092644. Wed Sep 03, 2014 10:30 am 


Moved here on request.
Just stuck this up on my blog so figured I'd share it here as well for those who haven't seen it before. Any geometry/settheory nerds around please feel free to point out all my mistakes haha!
You have a pea. Can you chop it up and rearrange the pieces (peaces...no?) to produce a ball the size of the sun? Our ordinary common sense would tell us no, down the rabbit hole of mathematics this is quite possible. Enter the BanachTarski paradox. This paradox is named after mathematicians Stefan Banach and Alfred Tarski who published their joint paper on paradoxical decomposition in 1924 (Just by the way...Stefan Banach used to write his maths problems and solutions directly onto his local cafe's tabletops, until his wife purchased a large notebook for him and left it with the cafe's owner).
What does the paradox state?
The BanachTarski paradox states that a three dimensional mathematical ball can be cut up into a finite number of pieces, which by rotating and translating the pieces but not changing their shape, can then reassembled to make two balls, each of which are the same size as the original ball. The corollary of the paradox is that we can take a ball, cut it into pieces, remove some of those pieces, and then reassemble the original ball. In cutting up the ball, we require at least 5 pieces in order for this to work. A stronger form of the paradox states that we can not just double a single ball into two balls, but reassemble any two solid objects into each other  for example a pea and the Sun. This violates our intuition as we appear to be able to increase the volume of an object without stretching anything or adding anything!
Now clearly this cannot be done in real life. It is worth clarifying and emphasising here that we are not talking about a physical ball, which we can bounce or flick or throw, but a mathematical ball, which is much less common sensical. Our savannahevolved intuitionion, accustomed to dealing with tangible medium sized physical things is obviously correct to tell us that you cannot physically chop up a pea and reassemble it into the Sun. We evolved to avoid predators and search for food  not to do abstract mathematics or quantum mechanics, so it is of little surprise that this paradox (or quantum mechanics) makes little intuitive sense. However it does make mathematical sense (for a stab at intuitive sense you have to imagine that the two balls are infinitely dense).
How does this work?
It essentially comes down to a definition of volume. When the ball is cut the pieces will form some sort of shapes. These shapes are exotic creatures and so do not have a defined volume, being more like a infinite scattering of points than a solid, and so by translating and rotating these shapes we can produce one ball from two. Using set theory speak, we decompose the ball into subsets and then put it back together in a different way. These subsets are known as nonmeasurable, meaning that they do not have a volume associated with them in the ordinary sense. The existence of non measurable sets depends on a controversial proposal in set theory called the axiom of choice.
The axiom of choice
The axiom of choice is an axiom in set theory that states that given a collection of sets there is a way to choose one element from each set. All of this discussion rests on the assumption that the axiom of choice is true, as the validity of the axiom allows for the existence of nonmeasurable sets  the pieces of illdefined volume that allow us to double the volume. Most mathematicians accept that the axiom of choice is true  however its repercussions (such as peas reassembling into Suns) has left some to adopt the philosophical rather than mathematical position that it cannot be true.





AspiringElf

1164119. Sun Dec 20, 2015 2:03 pm 


I'm not really competent to comment on this, but I did find it interesting. For one thing, I liked the phrase 'our savannahevolved intuition' (which is 'accustomed to dealing with tangible medium sized physical things').
PS: What you actually wrote was 'savannahevolved intuitionion' (sic). 




Spud McLaren

1164166. Sun Dec 20, 2015 5:35 pm 


AspiringElf wrote:  PS: What you actually wrote was 'savannahevolved intuitionion' (sic).  Yes. It's manylayered, with nothing at the centre. 




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